For $a, b>1$, $(a,b)=1$ iff $a$ does not divide $b$ and $b$ does not divide $a$?

Question

I am guessing the following statement is true:

For two integers $a$ and $b$ greater than $1$, $(a,b)=1$ if and only if $a$ does not divide $b$ and $b$ does not divide $a$.

Could anyone please confirm? Also if this statement is true, is it in the most general form relating the condition $(a,b)=1$ to non-divisibility of $a$ and $b$? In other words, is it true $a$ and $b$ must be integers greater than $1$?

Answer 1

Consider $a=2\cdot 3$ and $b=2\cdot 5$. Obviously $a\not| b$ and $b \not| a$ but $(a,b) = 2 \neq 1$.